Boost-invariant Hamiltonian approach to heavy quarkonia

Abstract

Light-front Hamiltonian formulation of QCD with only one flavor of quarks is used in its simplest approximate version to calculate masses and boost-invariant wave functions of $c\overline{c}$ or $b\overline{b}$ mesons. The quark-antiquark Hamiltonian is obtained in the lowest (second) order of a weak-coupling expansion scheme for Hamiltonians of effective particles in quantum field theory. The derivation involves a heuristic ansatz for a gluon mass-gap that is meant to account for non-Abelian color dynamics of virtual effective particles in the Fock components with one or more effective gluons and can be improved order by order in the expansion. Fortunately, the resulting quark-antiquark Hamiltonian does not depend on any details of the ansatz within a large class of possibilities. It is shown that in the Hamiltonian approach in its simplest version the strong coupling constant $\ensuremath{\alpha}$ and quark mass $m$ (for suitable values of the renormalization group parameter $\ensuremath{\lambda}$ that is used in the calculation), can be adjusted so that (a) masses of 12 lightest well-established $b\overline{b}$ mesons are reproduced with accuracy better than 0.5% for all of them, which means 50 MeV in a few worst cases and on the order of 10 MeV in other cases, or (b) masses of 11 lightest $c\overline{c}$ mesons are reproduced with accuracy better than 3% for all of them, which means better than 100 MeV in a few worst cases and on the order of 10 MeV in the other cases, while the parameters $\ensuremath{\alpha}$ and $m$ are near the values expected in the cases (a) and (b) by analogy with other approaches. A fourth-order study in the same Hamiltonian scheme will be required to explicitly include renormalization group running of the parameters $\ensuremath{\alpha}$ and $m$ from the scale set by masses of bosons $W$ and $Z$ down to the values of $\ensuremath{\lambda}$ that are suitable in the bound-state calculations. In principle, one can use the Hamiltonian approach to describe the structure, decay, production, and scattering of heavy quarkonia in all kinds of motion, including velocities arbitrarily close to the speed of light. This work is devoted exclusively to a pilot study of masses of the quarkonia in the simplest version of the approach.